## 1. Introduction

[2] IEs have been traditionally employed for the accurate solution of a wide variety of electromagnetic problems, such as the analysis of scatterers solved in the work of *Poggio and Miller* [1973], the analysis and design of boxed multilayered circuits described by *Eleftheriades et al.* [1996], and the efficient modal computation of arbitrarily shaped waveguides shown in the work of *Conciauro et al.* [1984]. These integral equation problems are usually solved following the well-known MOM described by *Harrington* [1968]. When such approach is applied to either electrically large structures, highly accurate applications, or objects with many small geometry details, matrix problems with a very large number of unknowns arise.

[3] To avoid the aforementioned drawback related to the IE-MOM approach, and take profit of the accuracy and efficiency advantages of such technique, the use of the wavelet transform has been recently proposed [see *Sarkar et al.*, 2002]. It was initially applied to microstrip circuits by *Wang and Pan* [1995] and *Sabetfakhri and Katehi* [1994], to the efficient waveguide mode computation by *Wagner et al.* [1993], and to the analysis of double-slot apertures in planar conducting screens by *Steinberg and Leviatan* [1993]. In the recent years, the application of the wavelet transform to the efficient analysis of 2-D electromagnetic scattering problems has become rather popular, as can be seen in the work of *Steinberg and Leviatan* [1994], *Wagner and Chew* [1995], *Baharav and Leviatan* [1996a, 1996b], *Nevels et al.* [1997], *Deng and Ling* [1999], *Baharav and Leviatan* [1998], *Golik* [2000], and *Guan et al.* [2000]. For instance, *Steinberg and Leviatan* [1994] proposed a periodic wavelet expansion to overcome the classical problem of expanding a given function of finite support. An extensive comparison of wavelets with other fast algorithms for solving the cited 2-D problem can be found in the work of *Wagner and Chew* [1995], where it is concluded that the use of wavelets in small problems is not very advantageous. *Baharav and Leviatan* [1996b], *Deng and Ling* [1999], and *Baharav and Leviatan* [1996a, 1998] proposed novel approaches for the incorporation of wavelet expansions into integral equations, which are mainly based on the impedance matrix compression using either adaptively constructed, or iteratively selected, wavelet basis functions. These wavelet transforms are computationally very expensive and they have not been selected for further study in this paper. Different wavelet bases for solving IEs were first studied in the work of *Nevels et al.* [1997], where semiorthogonal compact support bases outperformed infinite support orthogonal ones. However, *Golik* [2000] and *Guan et al.* [2000] studied the most suitable wavelet bases for scattering problems, and they conclude that, in such case, the orthogonality of wavelets is a crucial issue to be fulfilled. Another research direction is to use smooth local cosine bases instead of wavelet bases. This is proposed in the work of *Sweldens* [1996] and applied to scattering analysis in the work of *Pan et al.* [2003]. However, if no a priori information about the scatterer is known, many processing parameters should be adjusted in search for the best matrix compression and that increases consequently the computing cost.

[4] It is also convenient to remark that most of the works on scattering problems cited in the previous paragraph have been mainly focused on small 2-D scatterers, whose cylindrical conducting contours have been coarsely segmented. Such approach leads to small matrix problems, where the advantages of the wavelet approach are clearly lessened. Furthermore, the MOM solution followed in such references has been always formulated in the spatial domain, thus leading to incidence-dependent solutions, which are not very suitable for the later consideration of multiple scattering problems. In those methods, the solution of the scattering problem for different incidences involves the solution of a linear equation system for each incident field. In this paper, a formulation that computes the solution of the scattered field for any incidence solving only a limited set of linear equation systems is followed [see *Esteban et al.*, 1997].

[5] Despite of all the previously commented efforts, much research on the wavelet transform, and its practical application to the accurate solution of electromagnetic scattering problems, is still needed. Specially, wavelet-like bases have not been sufficiently studied in the frame of 2-D scattering problems compared to other popular wavelet bases such as Daubechies. Therefore, taking into account the present state-of-the-art reached in this research area, it seems appropriate to face the combination of wavelet and wavelet-like bases and the new spectral domain technique for the efficient and accurate solution of large MOM matrix problems with oscillatory kernels. If the application of wavelet theory in this new scenario gave good enough results, a further step for the very efficient and accurate solution of large scattering problems by personal computers without special memory arrangements would be given. The improvement of the efficiency related to the use of the wavelet transform for solving large 2-D scatterers would encourage its use in more complex 3-D scattering scenarios.

[6] This paper is organized as follows. First, the 2-D electromagnetic scattering problem under consideration is formulated and solved in the spectral domain. Next, the wavelet transform is successfully introduced for the compression of the corresponding impedance matrix. After testing several wavelet and wavelet-like families, the best ones have been selected to perform a comparative study included in this work. Such study has been realized through simple 2-D large scattering problems, such as circular and square cylinders, as well as strips. Very accurate numerical results for the scattering behavior of such objects are presented, and the efficiency gain provided by the use of wavelets is discussed. Finally, the main conclusions derived from the detailed study performed in this paper are briefly outlined.